The total surface area of a hemisphere is `1038.5 cm^(2)`. Find its diameter (in cm.).

Options

A. 21
B. `10.5`
C. 42
D. `31.5`

Answer

To find the diameter of a hemisphere given its total surface area, we can follow these steps:

### Step 1: Understand the formula for the total surface area of a hemisphere.
The total surface area (TSA) of a hemisphere is given by the formula:
\[ \text{TSA} = 3\pi r^2 \]
where \( r \) is the radius of the hemisphere.

### Step 2: Set up the equation using the given total surface area.
We know from the problem that the total surface area of the hemisphere is \( 1038.5 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 3\pi r^2 = 1038.5 \]

### Step 3: Solve for \( r^2 \).
To isolate \( r^2 \), divide both sides of the equation by \( 3\pi \):
\[ r^2 = \frac{1038.5}{3\pi} \]

### Step 4: Calculate \( r^2 \).
Using \( \pi \approx 3.14 \):
\[ r^2 = \frac{1038.5}{3 \times 3.14} \approx \frac{1038.5}{9.42} \approx 110.14 \]

### Step 5: Find the radius \( r \).
Now, take the square root of both sides to find \( r \):
\[ r = \sqrt{110.14} \approx 10.49 \, \text{cm} \]

### Step 6: Calculate the diameter \( d \).
The diameter \( d \) is twice the radius:
\[ d = 2r = 2 \times 10.49 \approx 20.98 \, \text{cm} \]

### Step 7: Round the diameter.
Rounding \( 20.98 \) to one decimal place gives us:
\[ d \approx 21 \, \text{cm} \]

### Final Answer:
The diameter of the hemisphere is approximately \( 21 \, \text{cm} \).

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